Circuit topology of self-interacting chains: implications for folding and unfolding dynamics

Cytosolic Interactome Protects Against Protein Unfolding in a Single Molecule Experiment

Single molecule techniques are particularly well suited for investigating the processes of protein folding and chaperone assistance. However, current assays provide only a limited perspective on the various ways in which the cellular environment can influence the folding pathway of a protein. In this study, a single molecule mechanical interrogation assay is developed and used to monitor protein unfolding and refolding within a cytosolic solution. This allows to test the cumulative topological effect of the cytoplasmic interactome on the folding process. The results reveal a stabilization against forced unfolding for partial folds, which are attributed to the protective effect of the cytoplasmic environment against unfolding and aggregation. This research opens the possibility of conducting single molecule molecular folding experiments in quasi-biological environments.

Folding kinetics of an entangled protein

The possibility of the protein backbone adopting lasso-like entangled motifs has attracted increasing attention. After discovering the surprising abundance of natively entangled protein domain structures, it was shown that misfolded entangled subpopulations might become thermosensitive or escape the homeostasis network just after translation. To investigate the role of entanglement in shaping folding kinetics, we introduce a novel indicator and analyze simulations of a coarse-grained, structure-based model for two small single-domain proteins. The model recapitulates the well-known two-state folding mechanism of a non-entangled SH3 domain. However, despite its small size, a natively entangled antifreeze RD1 protein displays a rich refolding behavior, populating two distinct kinetic intermediates: a short-lived, entangled, near-unfolded state and a longer-lived, non-entangled, near-native state. The former directs refolding along a fast pathway, whereas the latter is a kinetic trap, consistently with known experimental evidence of two different characteristic times. Upon trapping, the natively entangled loop folds without being threaded by the N-terminal residues. After trapping, the native entangled structure emerges by either backtracking to the unfolded state or threading through the already formed but not yet entangled loop. Along the fast pathway, trapping does not occur because the native contacts at the closure of the lasso-like loop fold after those involved in the N-terminal thread, confirming previous predictions. Despite this, entanglement may appear already in unfolded configurations. Remarkably, a longer-lived, near-native intermediate, with non-native entanglement properties, recalls what was observed in cotranslational folding.

A tile model of circuit topology for self-entangled biopolymers

Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and connected sums of these knots. This includes all proteins knots that have thus far been identified.

ProteinCT: An implementation of the protein circuit topology framework

The ability to describe the topology of a folded protein conformation is critically important for functional analysis, protein engineering, and drug design. Circuit topology is a unique topological framework which is widely applicable to protein analysis, yet a state-of-the art implementation of this concept is lacking. Here, we present an open-source Python-implemented circuit topology tool called ProteinCT. The platform provides a method for acquiring, visualizing, analyzing, and quantifying circuit topology data from proteins of interest. We mapped the universe of human proteins to a circuit topology space using conventional hardware within a few hours, demonstrating the performance of ProteinCT. In brief,•A Python-implemented circuit topology tool is developed to extract global and local topological information from a protein structure file.•Modules are developed to combine topological information with geometric and energetic information.•It is demonstrated that the method can be efficiently applied to a large set of proteins, opening a wide range of possibilities for structural proteomics research.

Circuit topology predicts pathogenicity of missense mutations

The contact topology of a protein determines important aspects of the folding process. The topological measure of contact order has been shown to be predictive of the rate of folding. Circuit topology is emerging as another fundamental descriptor of biomolecular structure, with predicted effects on the folding rate. We analyze the residue‐based circuit topological environments of 21 K mutations labeled as pathogenic or benign. Multiple statistical lines of reasoning support the conclusion that the number of contacts in two specific circuit topological arrangements, namely inverse parallel and cross relations, with contacts involving the mutated residue have discriminatory value in determining the pathogenicity of human variants. We investigate how results vary with residue type and according to whether the gene is essential. We further explore the relationship to a number of structural features and find that circuit topology provides nonredundant information on protein structures and pathogenicity of mutations. Results may have implications for the polymer physics of protein folding and suggest that “local” topological information, including residue‐based circuit topology and residue contact order, could be useful in improving state‐of‐the‐art machine learning algorithms for pathogenicity prediction.

Topological dynamics of an intrinsically disordered N‐terminal domain of the human androgen receptor

Human androgen receptor contains a large N‐terminal domain (AR‐NTD) that is highly dynamic and this poses a major challenge for experimental and computational analysis to decipher its conformation. Misfolding of the AR‐NTD is implicated in prostate cancer and Kennedy's disease, yet our knowledge of its structure is limited to primary sequence information of the chain and a few functionally important secondary structure motifs. Here, we employed an innovative combination of molecular dynamics simulations and circuit topology (CT) analysis to identify the tertiary structure of AR‐NTD. We found that the AR‐NTD adopts highly dynamic loopy conformations with two identifiable regions with distinct topological make‐up and dynamics. This consists of a N‐terminal region (NR, residues 1–224) and a C‐terminal region (CR, residues 225–538), which carries a dense core. Topological mapping of the dynamics reveals a traceable time‐scale dependent topological evolution. NR adopts different positioning with respect to the CR and forms a cleft that can partly enclose the hormone‐bound ligand‐binding domain (LBD) of the androgen receptor. Furthermore, our data suggest a model in which dynamic NR and CR compete for binding to the DNA‐binding domain of the receptor, thereby regulating the accessibility of its DNA‐binding site. Our approach allowed for the identification of a previously unknown regulatory binding site within the CR core, revealing the structural mechanisms of action of AR inhibitor EPI‐001, and paving the way for other drug discovery applications.

Circuit topology analysis of cellular genome reveals signature motifs, conformational heterogeneity, and scaling

Reciprocal regulation of genome topology and function is a fundamental and enduring puzzle in biology. The wealth of data provided by Hi-C libraries offers the opportunity to unravel this relationship. However, there is a need for a comprehensive theoretical framework in order to extract topological information for genome characterization and comparison. Here, we develop a toolbox for topological analysis based on Circuit Topology, allowing for the quantification of inter- and intracellular genomic heterogeneity, at various levels of fold complexity: pairwise contact arrangement, higher-order contact arrangement, and topological fractal dimension. Single-cell Hi-C data were analyzed and characterized based on topological content, revealing not only a strong multiscale heterogeneity but also highly conserved features such as a characteristic topological length scale and topological signature motifs in the genome. We propose that these motifs inform on the topological state of the nucleus and indicate the presence of active loop extrusion.

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Circuit topology quantifies heterogeneity in genomic arrangement

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Scale analysis reveals a characteristic length scale of 10 Mb in genome topology

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We identify highly conserved topological structures related to loop extrusion

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We suggest a topological model of chromatin arrangement for loop extrusion, the L-loop

Topological principles of protein folding

What is the topology of a protein and what governs protein folding to a specific topology? This is a fundamental question in biology. The protein folding reaction is a critically important cellular process, which is failing in many prevalent diseases. Understanding protein folding is also key to the design of new proteins for applications. However, our ability to predict the folding of a protein chain is quite limited and much is still unknown about the topological principles of folding. Current predictors of folding kinetics, including the contact order and size, present a limited predictive power, suggesting that these models are fundamentally incomplete. Here, we use a newly developed mathematical framework to define and extract the topology of a native protein conformation beyond knot theory, and investigate the relationship between native topology and folding kinetics in experimentally characterized proteins. We show that not only the folding rate, but also the mechanistic insight into folding mechanisms can be inferred from topological parameters. We identify basic topological features that speed up or slow down the folding process. The approach enabled the decomposition of protein 3D conformation into topologically independent elementary folding units, called circuits. The number of circuits correlates significantly with the folding rate, offering not only an efficient kinetic predictor, but also a tool for a deeper understanding of theoretical folding models. This study contributes to recent work that reveals the critical relevance of topology to protein folding with a new, contact-based, mathematically rigorous perspective. We show that topology can predict folding kinetics when geometry-based predictors like contact order and size fail.

Polynomial Invariant of Molecular Circuit Topology

The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.

Generalized Circuit Topology of Folded Linear Chains

A wide range of physical systems can be formally mapped to a linear chain of sorted objects. Upon introduction of intrachain interactions, such a chain can "fold" to elaborate topological structures, analogous to folded linear polymer systems. Two distinct chain-topology theories, knot theory and circuit topology, have separately provided insight into the structure, dynamics, and evolution of folded linear polymers such as proteins and genomic DNA. Knot theory, however, ignores intrachain interactions (contacts), whereas chain crossings are ignored in circuit topology. Thus, there is a need for a universal approach that can provide topological description of any folded linear chain. Here, we generalize circuit topology in order to grasp particularities typically addressed by knot theory. We develop a generic approach that is simple, mathematically rigorous, and practically useful for structural classification, analysis of structural dynamics, and engineering applications.

Circuit Topology Analysis of Polymer Folding Reactions

Circuit topology is emerging as a versatile measure to classify the internal structures of folded linear polymers such as proteins and nucleic acids. The topology framework can be applied to a wide range of problems, most notably molecular folding reactions that are central to biology and molecular engineering. In this Outlook, we discuss the state-of-the art of the technology and elaborate on the opportunities and challenges that lie ahead.

Folding Rate Optimization Promotes Frustrated Interactions in Entangled Protein Structures

Many native structures of proteins accomodate complex topological motifs such as knots, lassos, and other geometrical entanglements. How proteins can fold quickly even in the presence of such topological obstacles is a debated question in structural biology. Recently, the hypothesis that energetic frustration might be a mechanism to avoid topological frustration has been put forward based on the empirical observation that loops involved in entanglements are stabilized by weak interactions between amino-acids at their extrema. To verify this idea, we use a toy lattice model for the folding of proteins into two almost identical structures, one entangled and one not. As expected, the folding time is longer when random sequences folds into the entangled structure. This holds also under an evolutionary pressure simulated by optimizing the folding time. It turns out that optmized protein sequences in the entangled structure are in fact characterized by frustrated interactions at the closures of entangled loops. This phenomenon is much less enhanced in the control case where the entanglement is not present. Our findings, which are in agreement with experimental observations, corroborate the idea that an evolutionary pressure shapes the folding funnel to avoid topological and kinetic traps.

Mapping a single-molecule folding process onto a topological space

Physics of protein folding has been dominated by conceptual frameworks including the nucleation-propagation mechanism and the diffusion-collision model, and none address the topological properties of a chain during a folding process. Single-molecule interrogation of folded biomolecules has enabled real-time monitoring of folding processes at an unprecedented resolution. Despite these advances, the topology landscape has not been fully mapped for any chain. Using a novel circuit topology approach, we map the topology landscape of a model polymeric chain. Inspired by single-molecule mechanical interrogation studies, we restrained the ends of a chain and followed fold nucleation dynamics. We find that, before the nucleation, transient local entropic loops dominate. Although the nucleation length of globules is dependent on the cohesive interaction, the ultimate topological states of the collapsed polymer are largely independent of the interaction but depend on the speed of the folding process. After the nucleation, transient topological rearrangements are observed that converge to a steady-state, where the fold grows in a self-similar manner.

Topology of polymer chains under nanoscale confinement

Spatial confinement limits the conformational space accessible to biomolecules but the implications for bimolecular topology are not yet known. Folded linear biopolymers can be seen as molecular circuits formed by intramolecular contacts. The pairwise arrangement of intra-chain contacts can be categorized as parallel, series or cross, and has been identified as a topological property. Using molecular dynamics simulations, we determine the contact order distributions and topological circuits of short semi-flexible linear and ring polymer chains with a persistence length of l_p under a spherical confinement of radius R_c. At low values of l_p/R_c, the entropy of the linear chain leads to the formation of independent contacts along the chain and accordingly, increases the fraction of series topology with respect to other topologies. However, at high l_p/R_c, the fraction of cross and parallel topologies are enhanced in the chain topological circuits with cross becoming predominant. At an intermediate confining regime, we identify a critical value of l_p/R_c, at which all topological states have equal probability. Confinement thus equalizes the probability of more complex cross and parallel topologies to the level of the more simple, non-cooperative series topology. Moreover, our topology analysis reveals distinct behaviours for ring- and linear polymers under weak confinement; however, we find no difference between ring- and linear polymers under strong confinement. Under weak confinement, ring polymers adopt parallel and series topologies with equal likelihood, while linear polymers show a higher tendency for series arrangement. The radial distribution analysis of the topology reveals a non-uniform effect of confinement on the topology of polymer chains, thereby imposing more pronounced effects on the core region than on the confinement surface. Additionally, our results reveal that over a wide range of confining radii, loops arranged in parallel and cross topologies have nearly the same contact orders. Such degeneracy implies that the kinetics and transition rates between the topological states cannot be solely explained by contact order. We expect these findings to be of general importance in understanding chaperone assisted protein folding, chromosome architecture, and the evolution of molecular folds.

Topology of internally constrained polymer chains

An interacting partner can provide external control over folding rates and realized topologies.

On the role of topology in regulating transcriptional cascades

Topology of interactions in a transcriptional cascade determines the behavior of its signal-response profile and the activation states of genes.

Role of Proteome Physical Chemistry in Cell Behavior

We review how major cell behaviors, such as bacterial growth laws, are derived from the physical chemistry of the cell’s proteins. On one hand, cell actions depend on the individual biological functionalities of their many genes and proteins. On the other hand, the common physics among proteins can be as important as the unique biology that distinguishes them. For example, bacterial growth rates depend strongly on temperature. This dependence can be explained by the folding stabilities across a cell’s proteome. Such modeling explains how thermophilic and mesophilic organisms differ, and how oxidative damage of highly charged proteins can lead to unfolding and aggregation in aging cells. Cells have characteristic time scales. For example, E. coli can duplicate as fast as 2–3 times per hour. These time scales can be explained by protein dynamics (the rates of synthesis and degradation, folding, and diffusional transport). It rationalizes how bacterial growth is slowed down by added salt. In the same way that the behaviors of inanimate materials can be expressed in terms of the statistical distributions of atoms and molecules, some cell behaviors can be expressed in terms of distributions of protein properties, giving insights into the microscopic basis of growth laws in simple cells.

Designing the structure and folding pathway of modular topological bionanostructures

Polypeptides and polynucleotides are programmable natural polymers whose linear sequence can be easily designed and synthesized by the cellular transcription/translation machinery. Nature primarily uses proteins as the molecular machines and nucleic acids as the medium for the manipulation of heritable information. A protein's tertiary structure and function is defined by multiple cooperative weak long-range interactions that have been optimized through evolution. DNA nanotechnology uses orthogonal pairwise interacting modules of complementary nucleic acids as a strategy to construct defined complex 3D structures. A similar approach has recently been applied to protein design, using orthogonal dimerizing coiled-coil segments as interacting modules. When concatenated into a single polypeptide chain, they self-assemble into the 3D structure defined by the topology of interacting modules within the chain. This approach allows the construction of geometric polypeptide scaffolds, bypassing the folding problem of compact proteins by relying on decoupled pairwise interactions. However, the folding pathway still needs to be optimized in order to allow rapid self-assembly under physiological conditions. Again the modularity of designed topological structures can be used to define the rules that guide the folding pathway of long polymers, such as DNA, based on the stability and topology of connected building modules. This approach opens the way towards incorporation of designed foldamers in biological systems and their functionalization.

Topology sorting and characterization of folded polymers using nano-pores

Here we report on the translocation of folded polymers through nano-pores using molecular dynamic simulations. Two cases are studied: one in which a folded molecule unfolds upon passage and one in which the folding remains intact as the molecule passes through the nano-pore. The topology of a folded polymer chain is defined as the arrangement of the intramolecular contacts, known as circuit topology. In the case where intramolecular contacts remain intact, we show that the dynamics of passage through a nano-pore varies for molecules with differing topologies: a phenomenon that can be exploited to enrich certain topologies in mixtures. We find that the nano-pore allows reading of the topology for short chains. Moreover, when the passage is coupled with unfolding, the nano-pore enables discrimination between pure states, i.e., states in which the majority of contacts are arranged identically. In this case, as we show here, it is also possible to read the positions of the contact sites along a chain. Our results demonstrate the applicability of nano-pore technology to characterize and sort molecules based on their topology.

Design principles for rapid folding of knotted DNA nanostructures

Knots are some of the most remarkable topological features in nature. Self-assembly of knotted polymers without breaking or forming covalent bonds is challenging, as the chain needs to be threaded through previously formed loops in an exactly defined order. Here we describe principles to guide the folding of highly knotted single-chain DNA nanostructures as demonstrated on a nano-sized square pyramid. Folding of knots is encoded by the arrangement of modules of different stability based on derived topological and kinetic rules. Among DNA designs composed of the same modules and encoding the same topology, only the one with the folding pathway designed according to the ‘free-end' rule folds efficiently into the target structure. Besides high folding yield on slow annealing, this design also folds rapidly on temperature quenching and dilution from chemical denaturant. This strategy could be used to design folding of other knotted programmable polymers such as RNA or proteins.

Driven by complementary base pairing, artificial single-chain DNA is capable of forming complex 3D architectures if an appropriate folding pathway can be realised. Here, the authors describe the design principles for rapidly folding structures, exemplified through fabrication of a nanosized square pyramid.

Distance measures and evolution of polymer chains in their topological space

Conformational transitions are ubiquitous in biomolecular systems, have significant functional roles and are subject to evolutionary pressures. Here we provide a first theoretical framework for topological transition, i.e. conformational transitions that are associated with changes in molecular topology. For folded linear biomolecules, arrangement of intramolecular contacts is identified as a key topological property, termed as circuit topology. Distance measures are proposed as reaction coordinates to represent progress along a pathway from initial topology to final topology. Certain topological classes are shown to be more accessible from a random topology. We study dynamic stability and pathway degeneracy associated with a topological reaction and found that off-pathways might seriously hamper evolution to desired topologies. Finally we present an algorithm for estimating the number of intermediate topologies visited during a topological reaction. The results of this study are relevant to, among others, structural studies of RNA and proteins, analysis of topologically associated domains in chromosomes, and molecular evolution.

Circuit topology of linear polymers: a statistical mechanical treatment

Circuit topology landscapes of linear polymer chains with intra-chain contacts are defined and studied for their properties.